• How do I use the relationships among the sides in special right triangles?

5.1 Special Right Triangles

Theorem 5.1: 45o – 45o – 90o Triangle TheoremIn a 45o – 45o – 90o triangle, the hypotenuse is ____ times as long as each leg.

o45

o45

x

x

2x2x 2x 2c

222 cx c 22x 2x

2

_____leghypotenuse 2

5.1 Special Right Triangles

Example 1 Find lengths in a Find lengths in a 45o – 45o – 90o triangle

Solution

Find the value of x.

a.6

x45

a. By the Triangle Sum Theorem, the measure of the third angle must be ______. Then the triangle is a ____-____- 90o triangle, so by Theorem 5.1, the hypotenuse is ___ times as long as each leg.

4545 45

2_____leghypotenuse 2 ____-____- 90o

Triangle Theorem45 45

Substitute._____x 26

5.1 Special Right Triangles

Example 1 Find lengths in a Find lengths in a 45o – 45o – 90o triangle

SolutionFind the value of x.b. You know that each of the two

congruent angles in the triangle has a measure of ____ because the sum of the angle measure in a triangle is 180o.

45

_____leghypotenuse 2 ____-____- 90o Triangle Theorem

45 45

Substitute.________ x29

x

29b.

x

2

Divide each side by ___.2____x 2

22

29

Simplify.x____9

5.1 Special Right TrianglesCheckpoint. Find the value of Checkpoint. Find the value of xx. .

22

x1.45

222leghyp

x 22 222x

4x

5.1 Special Right TrianglesCheckpoint. Find the value of Checkpoint. Find the value of xx. .

2.x

212

x 2leghyp

212 x 222

12x

5.1 Special Right Triangles

Theorem 5.2: 30o – 60o – 90o Triangle TheoremIn a 30o – 60o – 90o triangle, the hypotenuse is ____ as long as the shorter leg, and the longer leg is ____ times as long as the shorter leg.

2

legshorter ___hypotenuse 2

o30

60

xx2

3x

___legshorter leglonger 3

3

5.1 Special Right Triangles

Example 2 Find the height of an equilateralFind the height of an equilateral triangle

Music You make a guitar pick that resembles an equilateral triangle with side lengths of 32 millimeters. What is the approximate height of the pick?

Draw the equilateral triangle described.Solution

A

B

C

32 32

32

Its altitude forms the longer leg of two ___-___- 90 triangles. The length h of the altitude is approximately the height of the pick. 16 16

30 60o60

___legshorter leglonger 3______h 16 3 mm_____ 7.27

5.1 Special Right Triangles

Example 3 Find lengths in a Find lengths in a 30o – 60o – 90o triangle

Find the values of x and y. Write your answer in simplest radical form.

Find the value of x.

Solution

___legshorter leglonger 3______ x8 3

x______

o30

o60y

8

x

Step 1

Substitute.

Divide each side by ___.3

83

Multiply numerator and denominator by ___.3x

______

______3

8

3

3

x______

Multiply fractions.3

38

5.1 Special Right Triangles

Example 3 Find lengths in a Find lengths in a 30o – 60o – 90o triangle

Find the values of x and y. Write your answer in simplest radical form.

Find the value of y.

Solution o30

o60y

8

x

Step 2

legshorter ___hypotenuse 2

_________y 2

3

38

________ 3

316

5.1 Special Right TrianglesCheckpoint. Find the value of the variable. Checkpoint. Find the value of the variable.

32

x

3.

30

60

___legshorter leglonger 3

x2x 3

32 3

6x

5.1 Special Right TrianglesCheckpoint. Find the value of the variable. Checkpoint. Find the value of the variable.

12h

4.

6

12

660

___legshorter leglonger 3

h 6 336h

5.1 Special Right Triangles

Pg. 166, 5.1 #1-25